Abstract
The behavior of all the solutions of the logistic equation with delay and diffusion in a sufficiently small positive neighborhood of the equilibrium state is studied. It is assumed that the Andronov–Hopf bifurcation conditions are met for the coefficients of the problem. Small perturbations of all coefficients are considered, including the delay coefficient and the coefficients of the boundary conditions. The conditions are studied when these perturbations depend on the spatial variable and when they are time-periodic functions. Equations on the central manifold are constructed as the main results. Their nonlocal dynamics determines the behavior of all the solutions of the original boundary value problem in a sufficiently small neighborhood of the equilibrium state. The ability to control the dynamics of the original problem using the phase change in the perturbing force is set. The numerical and analytical results regarding the dynamics of the system with parametric perturbation are obtained. The asymptotic formulas for the solutions of the original boundary value problem are given.
Highlights
The logistic equation with delay10.3390/math9131566u = r [1 − au(t − T )]uAcademic Editors: Juan RamónTorregrosa Sánchez, Alicia CorderoBarbero and Juan Carlos Cortés LópezReceived: 8 May 2021Accepted: 1 July 2021Published: 3 July 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations
We present the observation of the boundary value problem (3) and (4) dynamics in the neighborhood of the equilibrium state u0 for the coefficients d and a, which are close to some values d0 and a0, and the parameters r and T close to r0 and T0 for which the equality (7) holds
The main result is that according to the asymptotic formula (11), the nonlocal dynamics of the Equation (88) determines the local behavior of the boundary value problem (83)
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. We note that the zero solution in (1) is unstable, and the positive equilibrium state u0 = a−1 is asymptotically stable under the condition 0 < rT ≤ π2. Is of great importance for the stability of the equilibrium state u0 of the boundary value problem (3) and (4). We present the observation of the boundary value problem (3) and (4) dynamics in the neighborhood of the equilibrium state u0 for the coefficients d and a, which are close to some values d0 and a0 , and the parameters r and T close to r0 and T0 for which the equality (7) holds. The above results are well known [15,16,17] They describe the Andronov–Hopf bifurcation in regard to the boundary value problem (3) and (4).
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